On the K-theory of pullbacks

Abstract

To any pullback square of ring spectra we associate a new ring spectrum and use it to describe the failure of excision in algebraic $K$-theory. The construction of this new ring spectrum is categorical and hence allows to determine the failure of excision for any localizing invariant in place of $K$-theory. As immediate consequences we obtain an improved version of Suslin's excision result in $K$-theory, generalizations of results of Geisser and Hesselholt on torsion in (bi)relative $K$-groups, and a generalized version of pro-excision for $K$-theory. Furthermore, we show that any truncating invariant satisfies excision, nilinvariance, and cdh-descent. Examples of truncating invariants include the fibre of the cyclotomic trace, the fibre of the rational Goodwillie--Jones Chern character, periodic cyclic homology in characteristic zero, and homotopy $K$-theory. Various of the results we obtain have been known previously, though most of them in weaker forms and with less direct proofs.

arXiv commentv1: 33 pages, v2: 41 pages. Added a discussion of homotopy K-theory as truncating invariant; a sufficient criterion on a pullback square of ring spectra to give rise to a pullback square of categories of (perfect) modules; a new proof of the fundamental theorem; and a second appendix on cdh descent with coefficients. v3: 41 pages, published version

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